Holography and micro-holography of particle fields: A numerical standard

Optics Communications 285 (2012) 3013–3020

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Holography and micro-holography of particle fields: A numerical standard Xuecheng Wu a, Siegfried Meunier-Guttin-Cluzel b, Yingchun Wu a, Sawitree Saengkaew b, Denis Lebrun b, Marc Brunel b, Linghong Chen a, Sébastien Coetmellec b, Kefa Cen a, Gerard Grehan b,⁎ a b

State Key Laboratory of Clean Energy Utilization, Institute for Thermal Power Engineering, Zhejiang University, 38 Zheda Road, Hangzhou, 310027, China Department ‘Optique’, UMR 6614/CORIA, CNRS/Université et INSA de Rouen, BP 8, 76800 Saint Etienne du Rouvray, France

a r t i c l e

i n f o

Article history: Received 2 August 2011 Received in revised form 22 February 2012 Accepted 27 February 2012 Available online 12 March 2012 Keywords: Holography Digital holography Mie theory Computer holography

a b s t r a c t Holography is an ‘old’ technique for studying the behavior of clouds of droplets which finds a new interest with CCD cameras and real-time numerical reconstruction. Furthermore, the continued progress in camera characteristics (sensitivity, pixel number, digitalization level, and so on) opens the way to more accurate recording of the interference field. To gain a deep understanding of the technique, as well as an evaluation of the performance and limitations of digital holographic particle measurements under various conditions, standard holograms are required. In this paper, a general numerical standard of holograms of fields of particles based on rigorous near-field Lorenz–Mie scattering theory is presented. This theory makes possible the computation of holograms of fields of particles with an arbitrary number of particles of arbitrary size, arbitrary refractive index, arbitrary recording distance (near-field or far-field), and an arbitrary collecting angle (forward, off-axis, or backward scattering light). Several calculation examples are also given for the code validation and possible applications, including a new possible way to simultaneously measure the size, location, and refractive index of particles. © 2012 Elsevier B.V. All rights reserved.

1. Introduction In the field of multiphase flow, there has always been great interest in and demand for three dimensional (3D) transient measurement methods, among which holography is believed to be the most promising option. It has been used for a long time to study the behavior of sprays, gas-solid flows, liquid-bubble flows, and so on [1–3]. After a period of relative decline due to the complexities and time-consuming issues of both the processing of photographic plates and optical reconstruction, the technique finds a new interest with the use of computer vision systems, where a CCD camera is used to record the pattern of interference fringes and is combined with numerical reconstruction to extract the size and 3D location of particles [4]. However, up to now, digital holography is still not fully developed and several issues including better reconstruction algorithms [5,6], effective data processing of the reconstructed images [7,8], better depth resolution [9], better optical arrangement [10–15], simultaneous measurement of refractive index [16–19], and so on remain to be improved. To have the best command of digital holography, standard holograms of fields of particles are needed for evaluation of the numerical reconstruction algorithms, processing strategies of the reconstructed images, and effects of recording parameters as well as new technologies. The holograms used in

⁎ Corresponding author. Tel.: + 33 2 32 95 36 29. E-mail address: [email protected] (G. Grehan). 0030-4018/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2012.02.101

previous investigations, however, are produced either from simple diffraction theory [20], where only in-line holography with forward scattering is applicable, or from far-field Lorenz–Mie theory [21], where the particles are assumed to be far enough from the detector. The aim of this work is to introduce a fully numerical standard of holograms of fields of particles based on a more general theory, the near-field Lorenz–Mie Theory (LMT). 1 The use of the near-field LMT makes it possible to take into account the 3D shape of the particle, its complex refractive index, the near-field structure of the scattered light, the polarization of the laser, the interferences between light scattered by different particles, and so on. Furthermore, as the camera characteristics (pixel size and number of pixels) are also used to compute the simulated images, the physics of scattering and the camera digitalization are taken into account. From such simulated holograms, the performance of different processing strategies can be compared according to the recording configuration. The paper is organized in five sections. Section 2 is devoted to a brief recall of the theoretical background of the near-field LMT applied to holography. The assumptions inherent in the use of nearfield LMT are underlined. Section 3 introduces the algorithm used to simulate holograms from clouds of droplets. Section 4 compiles a series of exemplifying results and Section 5 presents the conclusion.

1

The code can be obtained by asking one of the authors directly.

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X. Wu et al. / Optics Communications 285 (2012) 3013–3020 s;j

Eφ ¼

2. Methodology The configuration under study is schematized in Fig. 1. A plane wave, called the incident wave, impinges on a cloud of N spherical particles. The ith particle has a location of (xi, yi, zi) in a Cartesian coordinate system OXYZ. A CCD detector, defined with a resolution of M × M and a pixel size of dpix, is located in an angular direction Ω at a distance R from the center of the Cartesian coordinate system. On this detector the interference between the light scattered by the particles and the reference beam (assumed to be a plane wave impinging perpendicularly on the detector) are recorded. Note that in the particular case of in-line Gabor holography the reference wave is the incident wave; for other configurations the reference beam can be arbitrarily defined. Under the assumption that multiple scattering can be neglected (no light scattered by more than one particle reaches the detector), this problem is N times the basic problem of the interference between an incident plane wave and the field scattered by one particle, which has already been essentially solved by Slimani et al. [22]. The rewritten solution of the Slimani et al. [22] solution to this more general problem is given below. This work is complementary to the one by Pu and Meng [23], which uses the far-field approximation, while here the distance between the particles and the detector is fully arbitrary. 2.1. The field scattered by one particle

s;j

∞     X −E0 nþ1 n 2n þ 1 ′ ½an ξ n kr j π n cosθj sinφj i ð−1Þ n ð n þ 1 Þ kr j n¼1     −ibn ξn krj τn cosθj

Hr ¼ E0

 1=2 ∞ h    i   X ε nþ1 n 2n þ 1 ″ 1 sinφj i ð−1Þ bn ξ n krj þ ξn krj P n cosθj μ nðn þ 1Þ n¼1

ð4Þ H s;j θ ¼

  ∞     X E0 ε 1=2 nþ1 n 2n þ 1 ′ ½−ian ξ n krj πn cosθj sinφj i ð−1Þ n ð n þ 1 Þ krj μ n¼1     þ bn ξn kr j τ n cosθj ð5Þ

H s;j φ ¼

  ∞     X E0 ε 1=2 nþ1 n 2n þ 1 ′ ½−ian ξ n krj τn cosθj cosφj i ð−1Þ n ð n þ 1 Þ krj μ n¼1     þbn ξn krj πn cosθj ð6Þ

where E0 is the amplitude of the incident plane wave and the Legendre functions πn and τn are defined by: 1

A Mie scatter center (a perfectly spherical, isotropic, and homogeneous particle with a diameter d and a complex refractive index m) is located at a point Oj of the Cartesian coordinate system OXYZ. The incident wave propagates from the negative Z to the positive Z. In the non-absorptive medium surrounding the particle, the scattered electromagnetic field components (denoted Vks, j, where V stands for E or H, s stands for scattered, j is the particle number, and k stands for the coordinate r, θ, or φ), in a direction θj, φj, and at a distance rj from the particle center are given by:

s;j

Er ¼ E0 cosφj

∞ X n¼1

nþ1

i

n

ð−1Þ

h    i   2n þ 1 ″ 1 a ξ kr j þ ξn kr j P n cosθj nðn þ 1Þ n n

πn ð cosθÞ ¼ P n ð cosθÞ= sinθ 1

τn ð cosθÞ ¼ dP n ð cosθÞ=dθ 1

P n ð cosθÞ ¼ − sin

dP n ð cosθÞ d cosθ

ξn ðkrÞ ¼ ψn ðkrÞ þ iχ n ðkrÞ

ð9Þ

ð10Þ

where ψn ðkrÞ ¼ krψn ðkrÞ ¼

ð2Þ

ð8Þ

The functions ξn(kr) are given by:

ð1Þ

Es;j θ

ð7Þ

where Pn(cosθ) are the classical Legendre polynomials.

ð1Þ ∞     X E nþ1 n 2n þ 1 ′ ½an ξ n krj τn cosθj ¼ 0 cosφj i ð−1Þ n ð n þ 1 Þ krj n¼1     −ibn ξn krj πn cosθj

ð3Þ

n

χ n ðkr Þ ¼ ð−1Þ

  πkr 1=2 J nþ1=2 ðkr Þ 2

  πkr 1=2 J −n−1=2 ðkr Þ 2

ð11Þ

ð12Þ

where Jn + 1/2(kr) are the classical half-order Bessel functions and k is the wavenumber 2π/λ. The scattering coefficients read ′

′

an ¼

ψn ðα Þψ n ðβÞ−mψ n ðα Þψn ðβÞ ξn ðα Þψ′ n ðβÞ−mξ′ n ðα Þψn ðβÞ

ð13Þ

bn ¼

mψn ðα Þψ′ n ðβÞ−ψ′ n ðα Þψn ðβÞ mξn ðα Þψ′ n ðβÞ−ξ′ n ðα Þψn ðβÞ

ð14Þ

where α = πd/λ is the size parameter of the particle under study and β = mα. The prime indicates the derivative of the function with respect to the argument for the value of the argument indicated between parentheses. 2.2. The reference field

Fig. 1. Configuration under study.

The reference beam, for on-axis as well as for off-axis configurations, is assumed to impinge perpendicularly on the detector. Then

X. Wu et al. / Optics Communications 285 (2012) 3013–3020

the six components of the incident field for an {xd, yd, zd} Cartesian coordinate system attached to the detector (where the coordinate zd is perpendicular to the detector surface) are given by: i

Exd ¼

E0 expð−ikRÞ A

ð15Þ

i

ð16Þ

i

ð17Þ

i

ð18Þ

Eyd ¼ 0 E zd ¼ 0 H xd ¼ 0 i

H yd ¼

E0 ðε=μ Þ1=2 expð−ikRÞ A

i

H zd ¼ 0

ð19Þ ð20Þ

where A is an attenuation factor which is equal to 1 for in-line holography but can be arbitrarily defined for off-axis configurations.

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particle alone while the second series corresponds to the cumulated holograms created respectively by 1, 2, 3, … , N particles collectively. To do the computations, two series of six matrices are used: one series is devoted to the computation of the holograms created by one particle alone while the second series is devoted to the computation of the cumulated holograms. In each series, one matrix is devoted to each of the field components. The main steps of the algorithm are: 1. The six components of the reference field are computed for each pixel of the detector and are saved in the two series of matrices. 2. The six components of the scattered field for one particle are computed for each pixel of the detector. These components are added to the two series of matrices previously introduced; before this step, the first series of matrices contains only the reference field while the second series (of matrices) contains the reference field summed with all the previously introduced particle contributions. 3. The components of the Poynting vector perpendicular to the detector are computed for each pixel of the detector and saved in a file. 4. A new particle is introduced. The first series of matrices is reinitialized to the six components of the reference field. The code loops to step 2 until the total number of particles under study (N) is reached.

2.3. The total field from several particles Assuming that multiple scattering can be neglected (the quantity of light scattered by more than one particle which impinges on the detector is negligible), the total field at the running point P located on the detector surface will be the sum of the reference field with all the scattered fields. Then the six components of the total field are given by: t

i

Vw ¼ Vw þ

N X

s;j

Uw

ð21Þ

j¼1

where N is the total number of particles in the control volume, V stands for E or H, w stands for X, Y, or Z in the Cartesian system associated to the detector, and U stands for the component E or H of the scattered field in the same Cartesian system. When the total field is known, the intensities (and the direction of propagation) are given by the Poynting vector: S¼

1 h t t i Re E ⋅H 2

ð22Þ

The intensity recorded by one pixel is directly proportional to the flux of S through its surface, that is to say to the component of S perpendicular to the detector. 3. Programming strategy A code has been developed to compute holograms for an arbitrary number of particles according to the theoretical background introduced above. The main characteristics of this code are: ➢ The properties (diameter and complex refractive index) as well as the 3D location of each particle can be freely defined. ➢ The detector can be located arbitrarily. Forward and sideward as well as rainbow or backward detection can be accurately simulated. The distance between the detector and the cloud of particles can be defined arbitrarily (near-field configurations as well as far-field configurations can be simulated). ➢ The amplitude of the reference beam (for off-axis configurations) is arbitrarily defined. To allow accurate analysis of the interactions between the light scattered by different particles, two series of files are created. The first series corresponds to the holograms produced by each individual

4. Application examples In this section, several cases, including holograms of particle fields in Gabor configuration, off-axis configuration, far-field recording, and near-field recording, are calculated by using the code. Numerical reconstruction of these holograms is also performed and analyzed. 4.1. Gabor configuration The first example corresponds to the simulation of an in-line Gabor configuration (as shown in Fig. 2). The light source is assumed to have a wavelength equal to 0.532 μm. The detector (512 × 512 pixels) is assumed to be located 10 cm from the centre of the coordinate system, collecting the light between − 2° and 2° (corresponding to a square detector with a side length of 6984 μm and a 13.67 μm step between two pixels). The computations are for water droplets (m = 1.3333–0.0i) with a diameter equal to 50 μm. The first droplet is located at {0 μm, 0 μm, 0 μm}, the second at {500 μm, 500 μm, 500 μm}, and the third at {− 500 μm, −500 μm, −500 μm}. For the first particle, the dimensionless Fraunhofer criterion (CF = πd 2/4λz) is equal to 0.034. At the bottom of Fig. 2, the interferences between the light scattered by the different particles are clearly visible. The top part of Fig. 3 displays three holograms corresponding to a pair of two 50 μm diameter water droplets separated by distances of 400, 1000, and 2000 μm, respectively, while the bottom part of Fig. 3 shows the corresponding intensity distributions along the central line. The effect of the lateral particle on the visibility of the fringe structure of the central one is quantified. 4.2. Off-axis configuration In off-axis configuration, particle holograms at several specific scattering angles (Brewster angle, 90°, rainbow angle, backward, etc.) have been investigated. Here the first two are given as examples. 4.2.1. 90° collection For the same particles (size and location) and the same Fraunhofer criterion, the holograms are now computed for a 90° off-axis configuration. The collection angle runs from 88° to 92°. The holograms for one and three particles displayed in Fig. 4 are characterized by several ghost images known to be due to the Moiré effect, which can be negligible in Gabor cases (see Fig. 2).

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X. Wu et al. / Optics Communications 285 (2012) 3013–3020

Fig. 2. Holograms of the Gabor configuration. The collection angle runs between −2° and 2°. First row: the last particle introduced alone. Second row: the simulated holograms created by all the particles introduced.

As we know, holograms of fields of particles typically look like a group of concentric circular rings. The origin of these interferences is the difference in optical path between a spherical wave (the scattered field) and a plane wave (the reference wave). The spatial frequency of those interference fringes increases with the distance from the ring center. As a CCD sensor with a limited resolution is used, the hologram of a field of particles is not always sufficiently sampled, especially for the outside part of the circular fringes as they have higher frequencies. An undersampling of the hologram may occur according to the Nyquist sampling criteria and gives rise to a Moiré pattern [24,25]. The Moiré patterns also exist for a forward scattering particle hologram but have a low visibility. As for the forward case, the scattered light is

1 particle 2 particles separated by 400 µm

15.0

1 particle 2 particles separated by 1000 µm

15.0

14.5

14.5

14.5

14.0

14.0

14.0

13.0 12.5

13.5

Intensity

13.5

Intensity

Intensity

15.0

dominated by diffraction which has a strong intensity in the center but drops very quickly (by several orders of magnitude), resulting in a rapid decrease in the intensity of the fringe pattern from the center to the outside and suppressing the Moiré effect. On the contrary, for off-axis scattering, the scattered light does not change too much and the amplitude of the interferences with the reference beam will be essentially constant. Then the visibility of the fringes in the hologram will be essentially constant on the detector surface, creating a strong Moiré effect. This can be reduced by decreasing the collection angle or increasing the number of pixels. Fig. 5 displays the holograms computed for the same configuration as for Fig. 4 except for the collection angle, which is now running from 89° to 91°.

13.0 12.5

13.5 13.0 12.5

12.0

12.0

12.0

11.5

11.5

11.5

11.0

11.0

0

100

200

300

Pixels

400

500

600

0

100

200

300

Pixels

400

500

600

1 particle 2 particles separated by 2000 µm

11.0

0

100 200 300 400 500 600

Pixels

Fig. 3. Behavior of the hologram for a pair of water droplets with 50 μm diameters located at {0, 0, 0} and {x, 0, 0}, where x is equal to 400, 1000, and 2000 μm respectively. The distance particles/detector is equal to 63 mm, corresponding to a collection angle running from − 2° to 2°.

X. Wu et al. / Optics Communications 285 (2012) 3013–3020

a) ϕ = 0°

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b) ϕ = 90°

Fig. 4. Holograms foe a 90° configuration (4° collection angle).

4.2.2. Brewster angle It is interesting to know the properties of holograms taken at the Brewster angle since we know that the reflection component is quite different between perpendicular and parallel polarizations. For particles with a refractive index equal to 1.5, Fig. 6 compares the holograms recorded in the near-field (2 mm from the particle, Fraunhofer's criterion is equal to 2.657). The particle diameter is equal to 60 μm. In Fig. 6a the polarization angle (φ) is equal to 0° while in Fig. 6b it is equal to 90°. The collection angle is running from 62° to 72°. In Fig. 6a, the hologram is characterized by a series of nearly perfect concentric rings, due to the interference between a plane wave (the reference beam) and a spherical wave (corresponding to light refracted through the particle). In Fig. 6b, this simple topology is lost. A second spherical wave is visible, characterized by the presence of a second system of concentric rings. This second spherical wave is created by the light externally reflected on the surface of the particle. Furthermore, the system of nearly vertical lines identifiable in Fig. 6b is the signature of the interferences between the reflected and refracted light scattered by the particle. To understand this difference of behavior the Debye theory can be used. Contrarily to the Lorenz–Mie theory, which combines all the contributions to the scattering field (diffraction, refraction, reflection, …), the Debye theory computes each contribution independently. Fig. 7 plots the intensity scattered in the Debye framework. For the two polarizations the refracted contribution (p = 1, dashed lines) is essentially the same while for the reflected contribution (p = 0, continuous lines) the contribution for φ = 0 (red line) is very small relative to both the reflected contribution for φ = 90° (blue line) and the refracted contributions, with a sharp minimum corresponding to the Brewster angle.

Fig. 6. Comparison between holograms recorded between 62° and 72° for a polarization equal to 0° or 90°. The difference between the two images is due to the Brewster angle effect (suppression of the reflected light).

This explanation can be tested by computing the holograms for the same configuration except for the value of the refractive index, which is equal to 1.5–2.0i, corresponding to opaque particles: no refracted light. As displayed in Fig. 8, for both polarization cases, only one rings system exists. The center of the rings system is located at the same position, corresponding to the rings created by reflection in Fig. 6b. Note that the central ring is white or black according to the polarization; this is due to the different change of phase at reflection as predicted by Fresnel laws. 4.3. Holograms of field of particles By using the code, it is also possible to simulate holograms created by a large number of particles. The size, refractive index, and location of each particle can be manually set one by one, or if the minimal and maximal values of particle size, refractive index, and location are defined, they can be produced randomly. In Fig. 9, two examples are given corresponding to holograms created by 20 and 100 droplets respectively. After reconstruction, the extracted 3D location and diameter can be compared to the initial data introduced to compute the holograms. 4.4. An example of validation of an inversion code 4.4.1. Sensitivity to distance and refractive index To illustrate a possible application of standard numerical holograms, some of them computed for different sizes, different refractive

Fig. 5. Holograms for a 90° configuration. The collection angle has been reduced to 2° (collection angle running from 89° to 91°), limiting the Moiré effect.

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X. Wu et al. / Optics Communications 285 (2012) 3013–3020

1e+6

Scattered Intensity

1e+5 1e+4 1e+3 1e+2 1e+1 p = 0, ϕ = 0° p = 0, ϕ = 90° p = 1, ϕ = 0° p = 1, ϕ = 90°

1e+0 1e-1 1e-2 40

60

80

100

Fig. 9. Holograms for A) 20 particles arbitrarily located in (− 1500 μm b x b 1500 μm, − 1500 μm b y b 1500 μm, − 200 μm b z b 200 μm, 75 μm b d b 76 μm and a refractive index equal to 1.333) and B) 100 particles (same range for locations but diameters randomly selected between 1 and 100 μm). The collecting angle runs from − 2° to 2°.

Scattering angle Fig. 7. The reflected (p = 0) and refracted (p = 1) contributions computed in the Debye framework for the two polarizations.

index values, different pixel steps in the recording plane, and different distances between the particle and the detector have been processed by using an inversion scheme based on the wavelet transform (see references [26–28] for example). Tables 1 and 2 compile the results for a Fraunhofer criterion equal to 0.03 and 0.11 respectively. In these two tables, the results (particle size and longitudinal location) are not sensitive to the value of the refractive index. On the contrary the longitudinal location and the particle size are sensitive to the value of the Fraunhofer criterion. The smaller the Fraunhofer criterion is, the better the measurements are. In Table 1, the accuracy of the longitudinal measurement is equal to about 0.8% for the 4 μm particle and 0.4% for the 32.0 μm particle. With regard to size, errors are larger: 3.4% and 2.8%. In Table 2, for a larger Fraunhofer criterion, the measurements errors are larger: 3.8% and 4.6% for the longitudinal location and 33% and 33% for the size. Alternatively Fig. 10 plots the relative error in the diameter and the depth position versus the refractive index value. Fig. 10a is for a 4 μm particle located 200 μm (four times closer than for Table 1) from the detector with an equivalent pixel step of 0.2 μm while Fig. 10b is for a 32 μm particle located 12800 μm (again nearly four times closer than for Table 1) from the detector and with an equivalent pixel step of 1.2 μm. In Fig. 10a, for 4 μm particles, errors are smaller than 3%, independently of the value of the real part of the refractive index, while for transparent particles, the errors are significantly larger (up to 15%). On the contrary, in Fig. 10b, for 32 μm particles, the error is small and independent of the refractive index value for both the opaque and transparent particles.

By computing such holograms we were able to evaluate reconstruction methods and associated algorithms before conducting any experiments. 4.5. Near-field reconstruction In this section, the reconstructed images of a near-field off-axis scattering in-line recorded hologram of one particle are investigated. Fig. 11 shows the computed hologram (dp = 60 μm, m = 1.5–0i, θ = 70° with a recording distance from the hologram equal to R = 1 mm, view A) and the corresponding reconstructed images (views B and C). It is shown in Fig. 11B that for a transparent particle two glare spots corresponding to the refracted light spot (left) and reflected light spot (right) are clearly reconstructed at a depth close to the particle center (reconstructed depth position: 963 μm). Fig. 11C shows the reconstructed image along depth directions (from 750 μm to 1250 μm), which indicates that the scattered light spots have a large depth of focus which seems to be similar to the case for an in-line scattering hologram of particles. But the point is that the 3D locations and their intensity ratio of the two glare spots may be determined with a certain accuracy by means of digital imaging processing, which opens a possible way to simultaneously measure the 3D location, size, and refractive index (both real and imaginary parts) for sufficiently transparent spherical particles. For totally opaque particles, however, it does not seem possible to obtain even the center location and size of the particle since only the location of the reflected light spot can be obtained. 5. Conclusion A code has been developed in the framework of the near-field Lorenz–Mie theory to predict the holograms created by an ensemble

Table 1 Comparisons between the original parameters (diameter and distance) and the measurements for CF = 0.03. Pixel step is 4 μm. Original parameters

Measurements

Diameter (μm) Refractive index Distance (μm) Diameter (μm) Distance (μm)

Fig. 8. Hologram computed for the same configuration as in Fig. 6 but with the refractive index equal to 1.5–2.0 I (only reflected contribution exists).

4.0 4.0 4.0 4.0 32.0 32.0 32.0 32.0

1.3 1.4 1.5 1.5–1.0i 1.3 1.4 1.5 1.5–1.0i

800 800 800 800 50000 50000 50000 50000

3.87 3.87 3.85 3.86 33.1 33.1 33.1 33.1

810.8 789.8 797.8 799.9 49802 49802 49813 49802

X. Wu et al. / Optics Communications 285 (2012) 3013–3020

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Table 2 Comparisons between the original parameters (diameter and distance) and the measurements for CF = 0.11. Pixel step is 4 μm. Original parameters

Measurements

Diameter (μm) Refractive Index Distance (μm) Diameter (μm) Distance (μm) 4.0 8.0 16.0 32.0 4.0 8.0 16.0 32.0 4.0 8.0 16 32

1.3 1.3 1.3 1.3 1.4 1.4 1.4 1.4 1.5 1.5 1.5 1.5

200 800 3200 12800 200 800 3200 12800 200 800 3200 12800

2.18 5.08 10.92 22.56 2.61 5.02 10.04 20 3.21 5.26 10.72 21.28

213.56 835.74 3053.7 12186.2 191.14 767.07 3051.7 12180 199.15 836.74 3335.5 13355.4

of perfectly spherical, homogeneous, and isotropic particles located at an arbitrary distance from the detector. The recording of the scattered field and of its interferences with a reference field (assumed to be a plane wave) by a CCD camera is taken into account, including the Moiré effect. As the computed holograms depend directly on the main parameters of the physics of scattering (size, 3D-shape, complex refractive index, polarization, finite distance from the detector, etc.) and of the camera (pixel size and number), they are a natural candidate to be used as standard to validate different procedures for extracting information from the fringe patterns (3D location, size,

Relative errors for location and size, %

a 20

Depth position Error for Transparent particle Size Error for Transparent particle depth position Error for Opaque particle Size Error for Opaque particle

Fig. 11. A typical near-field hologram and the associated reconstructed images of an off-axis scattering particle hologram (dp =60 μm, m=1.5–0i, R=1 mm, 512×512, dpix =0.7 μm), corresponding to a collecting angle running from 60° to 80°. a) The hologram; B) image at depth position Z=963 μm; c) image along depth direction (from 750 μm to 1250 μm).

15

10

5

0 1.3

1.4

1.5

1.6

Refractive index Relative errors for location and size, %

b

refractive index) and compare their capabilities in terms of accuracy and computation time. Some numerical holograms have been presented, including nearfield holograms, exemplifying their use as standards. Experimental application of the program will be performed in the future. This approach can be easily adapted to other theories (such as Debye theory to explain some behaviors), other shapes of particles (multilayered particles, cylinder, spheroid, etc.), and other shapes of beam (such as circular Gaussian beam, elliptical Gaussian beam, and arbitrary beam, including femtosecond pulses).

6 Depth position Error for Transparent particle Size Error for Transparent particle

Acknowledgements The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (NSFC) projects (grant 50806067), the National Basic Research Program of China (grant 2009CB219802), the Program of Introducing Talents of Discipline to University (B08026), the European program INTERREG IVa-C5: Cross-Channel Center for Low Carbon Combustion.

4

2

References depth position Error for Opaque particle Size Error for Opaque particle 0 1.3

1.4

1.5

1.6

Refractive index Fig. 10. Relative error in size and depth position for particles versus refractive index value (a: dp = 4 μm, z = 200 μm; b: dp = 32 μm, z = 12800 μm).

[1] [2] [3] [4]

Y. Pu, X. Song, H. Meng, Experiments in Fluids 29 (2000) S117. Y. Pu, H. Meng, Experiments in Fluids 29 (2000) 184. H. Meng, F. Hussain, Fluid Dynamics Research 8 (1991) 33. H. Meng, G. Pan, Y. Pu, S.H. Woodward, Measurement Science and Technology 15 (2004) 673. [5] C.G. Liu, D.Y. Wang, Y.Z. Zhang, Optical Engineering 48 (2009) 105802. [6] A. Sharma, G. Sheoran, Z.A. Jaffery, Moinuddin, Optics and Lasers in Engineering 46 (2008) 42. [7] S.A. Wormald, J. Coupland, Applied Optics 48 (2009) 6400.

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[8] G. Pan, H. Meng, Applied Optics 42 (2003) 827. [9] W. Yang, A.B. Kostinski, R.A. Shaw, Optics Letters 30 (2005) 1. [10] W.J. Zhou, Q.S. Xu, Y.J. Yu, A. Asundi, Optics and Lasers in Engineering 47 (2009) 896. [11] D.S. Monaghan, D.P. Kelly, N. Pandey, B.M. Hennelly, Optics Letters 34 (2009) 3610. [12] L. Granero, V. Mico, Z. Zalevsky, J. Garcia, Optics Express 17 (2009) 15008. [13] J. de Jong, J.P.L.C. Salazar, S.H. Woodward, L.R. Collins, H. Meng, International Journal of Multiphase Flow 36 (2010) 324. [14] L. Cao, G. Pan, J. de Jong, S. Woodward, H. Meng, Applied Optics 47 (2008) 4501. [15] C.H. Atkinson, J. Soria, 16th Australasian Fluid Mechanics Conference (Crown Plaza, Gold Coast, Australia, 2007), 2007. [16] S.-H. Lee, Y. Roichman, G.-R. Yi, S.-H. Kim, S.-M. Yang, A. Blaaderen, P. Oostrum, D.G. Grier, Optics Express 15 (2007) 18275. [17] C. Fook Chiong, G.G. David, Digital Holography and Three-Dimensional Imaging, OSA Technical Digest (CD), Optical Society of America, 2009, PJTuB36. [18] S. Meunier-Guttin-Cluzel, X.C. Wu, G. Grehan, K.F. Cen, Congrès Francophone de Techniques Laser, CFTL 2010 (Vandoeuvre-lès-Nancy, France), 2010. [19] S. Meunier-Guttin-Cluzel, G. Gréhan, X.C. Wu, L.H. Chen, K.Z. Qiu, K.F. Cen, Asian Aerosol Conference AAC09 (Bangkok, Thailand, 2009), 2009.

[20] Y. Zhang, G. Shen, A. Schroder, J. Kompenhans, Optical Engineering 45 (2006) 075801. [21] G. Pan, "Digital holographic imaging for 3D particle and flow measurements," PhD (State University of New York at Buffalo, Buffalo, 2003). [22] F. Slimani, G. Grehan, G. Gouesbet, D. Allano, Applied Optics 23 (1984) 4140. [23] Y. Pu, H. Meng, Journal of the Optical Society of America a – Optics Image Science and Vision 20 (2003) 1920. [24] X.C. Wu, G. Gréhan, S. Meunier-Guttin-Cluzel, Y.C. Wu, L.H. Chen, H. Zhou, K.Z. Qiu, K.F. Cen, 15th Int. Symp. on Applications of Laser Techniques to Fluid Mechanics, 2010. [25] D. Lebrun, C. Ozkul, D. Allano, A. Leduc, Journal of Optics 22 (1991) 175. [26] C. Burage-Lefebvre, S. Coëtmellec, D. Lebrun, C. Özkul, Optics and Lasers in Engineering 33 (2000) 409. [27] S.L. Pu, D. Allano, B. Patte-Rouland, M. Malek, D. Lebrun, K.F. Cen, Experiments in Fluids 39 (2005) 1. [28] S.L. Pu, "Développement de méthodes interférométriques pour la caractérisation des champs de particules.," PhD (Rouen, France, and Hangzhou, Chine, 2005).